====== Convolution ====== [[http://24.229.196.141:81/|Not that convolution...]] ===== Definition ===== Discrete time convolution: $$ (f \ast g)[n] \equiv \sum_{m = -\infty}^{\infty} f[m]g[n-m] $$ Continuous time convolution: $$ (f \ast g)(t) \equiv \int_{-\infty}^{\infty} f(\tau)g(t - \tau) d\tau $$ ===== Properties ===== * Convolution is commutative: $$ f \ast g = g \ast f $$ * Convolution is associative: $$ f \ast (g \ast h) = (f \ast g) \ast h $$ * Convolution is distributive over addition: $$ (f + g) \ast x = f \ast x + g \ast x $$ ===== Frequency domain ===== * Convolution in the time/spatial domain is equivalent to multiplication in the frequency domain. The inverse is also true.